The most common method for diagnosis of cardiac electrophysiological processes routinely used in clinical practice is electrocardiography in 12 standard leads. Simplicity and low cost of the standard electrocardiographical study together with its relatively high informativity have lead to its extremely widespread use in the daily practice.
However, the electrocardiographical method has principled limitations. Activity of certain compartments of the myocardium is inadequately reflected in electrocardiographical signals registered in standard leads. As an example, difficulties in ECG-diagnosis of myocardial infarction of back-basal compartments of the left ventricle may be named. Furthermore, according to the superposition principle of electrodynamics, any electrocardiogram is the sum of electric potentials coming from sources at a great number of myocardium points. Since electrophysiological processes in different areas of the cardiac muscle proceed simultaneously, it is rather difficult to determine a local electric activity of the myocardium on standard ECG-leads. For example, an atrial re-polarization wave in humans in conditions of a normal cardiac rhythm is not revealed in ECG, as it is “hidden” by a high-amplitude QRS-complex reflecting a ventricular depolarization. The vector-electrocardiography technique is characterized by the same limitations.
Greater possibilities are provided by a method for surface electrocardiographical mapping of the chest. The method consists in a synchronic registration of multiple (from 40 to 250 and more) unipolar ECG-leads from the chest surface and in constructing maps of the electric potential distribution over the chest surface by interpolation for each discrete moment of the cardiocycle.
However, this method does not allow one to determine more precisely a local electric activity of the myocardium. If an electrode is located on the chest surface, contributions to ECG-signal from the nearest and most remote, regarding a registration electrode, segments of the myocardium differ from each other by approximately one order. For an electrode placed on the heart surface this difference is three orders. In this connection, for revealing a local electric activity of the heart, methods of invasive ECG registration are used with an attempt to bring electrodes closely to the heart surface as much as possible.
Transesophageal electrophysiological study of the heart is based on inserting a probe with registration electrodes into the esophagus cavity. The esophagus at its certain part adjoins close enough to posterior wall of the left atrium and to posterior wall of the left ventricle; therefore, intraesophageal ECG-signals selectively register the activity of these heart compartments. Intraesophageal electrocardiography is applied, in particular, for differential diagnosis of supraventricular and ventricular arrhythmias (Transesophageal electrostimulation of the heart (Under edit. Sulimov V. A., Makolkin V. I.). Moscow: Meditsina, 2001.—208 pp. [in Russian]).
For the same purposes in conditions of reanimation and intensive therapy departments, ECG-registration from the vena supra-cava via postclavicular catheter is used. The role of an electrode plays the column of saline solution within catheter. According to this technique, an activity of the right atrium is mainly registered (Kalinin V. V. Application of ECG recorded through postclavicular catheter for differential diagnosis of supraventricular arrhythmias.—Proceedings of V Session of MNOAR. Moscow: 2005 [in Russian]).
However, methods above-mentioned permit one to reveal a local electric activity only of individual structures of the heart.
For a complex evaluation of cardiac electrophysiological processes and topical diagnosis of cardiac rhythm disturbances, an invasive electrophysiological study of the heart based on the direct registration of a set of electrocardiograms from epicardial or endocardial surfaces of the heart is carried out. These methods may be applied on “open-heart” in conditions of thoracotomy, as well as on the basis of intervention technologies of inserting registration devices (catheters) into cardiac cavities by transvascular access or into pericardial cavity by its fluoroscopically-guided transskin puncture.
Up-to-date realizations of methods above-mentioned are directed to a precise determination of three-dimensional (3-D) coordinates of registration electrodes by non-fluoroscopic techniques and to a visualization of results in the form of isopotential and isochronous maps on models of heart compartments with means of computer graphics. Computer models of heart compartments are constructed at a great number of electrogram-registration points with known coordinates, as well as on the basis of CT or MRT data of the heart (Revishvili A. Sh., Rzaev F. G., Djetybaeva S. K. Electrophysiological diagnosis and intervention treatment of complicated forms of heart rhythm disturbances with using a system of three-dimensional electro-anatomical mapping.—Vestn. Aritmol. 2004, 34: 32-37 [in Russian]; Pokushalov E. A., Turov A. N., Shugaev P. L., Artemenko S. L. Radiofrequency ablation of ventricular tachycardia by transpericardial approach.—Vestn. Aritmol. 2006, 44: 58-62 [in Russian]).
To this group of methods, one can also refer methods for non-contact endocardial mapping based on inserting a “swimming” balloon catheter into cardiac cavities, registering a set of electrograms on the heart surface and reconstructing endocardial electrograms by computational way on data obtained (Schilling R. J., Kadish A. H., Peters N. S. et al. Endocardial mapping of atrial fibrillation in the human right atrium using a non-contact catheter.—European Heart Journal. 2000, 21: 550-564).
The drawback of methods above-disclosed which is avoided in the present invention is their invasive character.
Analogues of the present invention are methods for electrogram reconstructing at internal points of the chest by computational way according to the data of synchronic registration of ECG sets on the chest surface.
These methods are based on solution of the inverse problem of electrocardiography. The statement of the inverse problem of electrocardiography (IP ECG) is formulated in works of Barr D., Spach M. Solutions of the inverse problem directly expressed in terms of potentials//Theoretical fundamentals of electrocardiology [Russian translation under edit. Nelson K. V. and Geselovitz D. V.]—Moscow: Meditsina 1979, pp. 341-352; MacLeod R. S., Brooks D. H. Recent progress in the inverse problem in electrocardiology//IEEE Eng. in Med. Bio. Mag. 17:1, pp. 78-83, January 1998; Rudy Y., Messinger-Rapport B. J. The inverse problem in electrocardiography: Solutions in terms of epicardial potentials. CRC Crit. Rev. Biomed. Eng. 1988, 16: 216-268.
From the mathematical standpoint, IP ECG is a problem of harmonic continuation of the potential in the direction of sources, i.e., the Cauchy problem for the Laplace equation. Computational domain, in which the Laplace equation is defined, represents a part of the chest bounded by heart external surface, chest surface on which ECG-registration is accessible, and by imaginary cross-sections of the chest at the level of the diaphragm and clavicles.
At that part of the chest surface where ECG-registration is accessible, values of the electric potential obtained as a result of ECG-mapping, as well as the condition of equality-to-zero of a potential normal derivative are given. These data compose the Cauchy conditions.
The Cauchy problem consists in finding the electric field potential in indicated domain and its trace on the heart surface and on cross-sections of the chest in such a way that the potential in computational domain would satisfy the Laplace equation, while on the torso surface where ECG-registration is accessible it would satisfy the Cauchy conditions.
According to Hadamard, the Cauchy problem for the Laplace equation is ill-posed, as any negligible errors in the condition may result in arbitrary large errors in the solution. For solving the Cauchy problem for the Laplace equation, it is necessary to apply special so-called regularizing algorithms of solution (Denisov A. M. Introduction to the theory of inverse problems [in Russian].—Moscow: Moscow State University, 1994; Tikhonov A. N., Arsenin V. Ya. Methods for solution of incorrect problems [in Russian].—Moscow: Nauka, 1986, 312 pp.).
To solve the Cauchy problem for the Laplace equation in above-disclosed statement (the inverse problem of electrocardiography) by an analytical way appears to be impossible. Therefore, the inverse problem of electrocardiography is numerically solved by means of computational mathematics with using computer techniques.
The specific way to solve the inverse problem of electrocardiography, besides aspects associated with surface ECG-mapping, defines                a method for determination and representation in a “numerical form” of heart and torso boundary surfaces;        an algorithm of numerical solution of the problem.        
One of the ways for solving the inverse problem of electrocardiography is a method for reconstructing the electric field on “quasi-epicard”, i.e., on a conditional spherical surface surrounding the heart. From the mathematical standpoint, this method is based on representation of the heart electric field potential in the form of a harmonic polynomial (sphere function) whose coefficients are found from the condition of equality (or the minimum of mean square deviation) of values of polynomial and values of an ECG-signal at points of its registration with taking into account the equality-to-zero of a potential normal derivative on the chest surface. For providing the stability of solution, polynomial of degree not higher than 4 is used. The essential disadvantage of this method is that, when the radius of sphere diminishes, i.e., as “quasi-epicard” surface approximates to a real surface of the heart, the accuracy of potential reconstructing sharply drops. When “quasi-epicard” surface approximates to the chest surface, the resolution of the method in terms of revealing a local electric activity of the myocardium decreases (Titomir L. I., Kneppo P. Mathematical modeling of heart's bioelectric generator.—Moscow: Nauka, Physmathlit, 1999.—448 pp. [in Russian]; Titomir L. I., Trunov V. G., Aidu E. A. I. Noninvasive electrocardiography.—Moscow: Nauka, 2003.—198 pp. [in Russian]).
In order to solve boundary problems for the Laplace equation, methods of integral equations of the potential theory, more known in English-written literature as boundary element methods, are widely used (Brebbia C., Telles J., Wrobel L. Boundary element methods [Russian translation].—Moscow, Mir, 1987). The present approach to IP ECG solution is proposed in works of Taccardi E., Plonzi R., Barr R. (Barr R., Spach M Inverse problem solutions directly expressed in terms of a potential//Theoretical fundamentals of electrocardiography [Russian translation]. The above-mentioned methods suppose, in particular, the representation of heart and torso surfaces as polygonal surfaces, i.e., splitting boundary surfaces into a great number of triangular elements. According to the boundary element method, IP ECG for a homogeneous model of the chest is reduced to solving a system of two Fredholm integral equations of 1st and 2nd kinds, which is approximately substituted by a system of matrix-vector equations:A11x+A12y=c1,A11x+A22y=c2  (1)where Ai,j are known matrices; x1,x2 are unknown vectors having a sense of sought-for values of the potential and its normal derivatives in nodes of triangulation grids approximating the heart surface and torso cross-section surfaces; c1, c2 are known vectors calculated on known data of the problem.
In the method of noninvasive epicardial mapping suggested by Shakin V. V. et al. the following algorithm of IP ECG solution was used.
The system of matrix-vector equations (1) by means of elementary transformations was reduced to a system of linear algebraic equations (SLAE) to be resolved in explicit form:ΦH=ZHB·ΦB,  (2)where ΦH is an unknown vector having a sense of sought-for values of the potential and its normal derivatives in nodes of triangulation grids approximating the heart surface and torso cross-section surfaces; ZHB is a known matrix; ΦB is a known vector. For computing matrix ZHB, it is necessary to use an inversion procedure of matrices entering the system (1), one of matrices to be inversed being non-square and ill-conditioned. For implementation of this procedure, constructing a Moore-Penrose pseudo-inverse matrix on the basis of SVD-decomposition of an initial matrix and substituting small singular values by zeroes were performed.
Heart and torso surfaces were represented as simplified models in the form of cylindrical and ellipsoidal surfaces to be constructed on the basis of two-projection roentgenography of the chest. Results of mapping in the form of isopotential and isochronous maps were imposed on model scanned-schemes of heart compartments. This methodology was used for revealing a localization of additional pathways (APW) at manifested WPW syndrome and ectopic sources at ventricular extrasystole (Shakin V. V. Computational electrocardiography [in Russian].—Moscow: Nauka, 1980).
In his works, Shakin V. V. has emphasized a promising outlook of the application of computed tomography techniques for more precise constructing the torso and heart surfaces; however, this approach could not be used because of unsatisfactory development of methods for computer tomography of the heart.
The most similar to a method claimed here (prototype) is the methodology of noninvasive electrocardiographic imaging (ECGI).
In this methodology, a surface mapping is performed with using 240 unipolar electrodes placed in a special vest to be put on a patient for a study period. The torso and heart surfaces are determined on the basis of computer (CT) or magneto-resonance (MRT) tomography of the chest. A reconstruction algorithm is based on solution of the inverse problem of electrocardiography with using the boundary element method.
The torso and heart surfaces are approximately represented as polygonal surfaces. For solving IP ECG, the system of matrix-vector equations (1) is also used, which is reduced to a system of linear algebraic equations by elementary transformationsAx=c  (3)where x is an unknown vector having a sense of sought-for values of the potential in nodes of triangulation grids approximating the heart surface and torso cross-section surfaces; A is a known matrix; c is a known vector.
The system of linear algebraic equations (3) is ill-conditioned. For its solving the Tikhonov regularization method and the iterative regularization method based on GMRes-algorithm are applied. The Tikhonov method is based on solution of the following system instead of the system (3):(AT·A+αE)x=ATc, where AT is a matrix transponated in respect of matrix A; E is a unit matrix; α is a regularization parameter (a small positive real number).
The iterative regularization method is based on solution of the system (3) by a method of sequential approximations with restricting a number of iterations on the basis of GMRes-algorithm; this method belongs to a group of Krylov subspace methods (Ramanathan C., Ghanem R. N., Jia P., Ryu K., Rudy Y. Electrocardiographic Imaging (ECGI): A Noninvasive Imaging Modality for Cardiac Electrophysiology and Arrhythmia//Nature Medicine, 2004; 10: 422-428; Rudy Y., Ramanathan, C., R. N. Ghanem, R. N., Jia P. System and Method For Noninvasive Electrocardiographic Imaging (ECGI) Using Generalized Minimum Residual (GMRes)//U.S. Pat. No. 7,016,719 B2, 2006).
The similar technique was used in works of Berger T, Fisher G., Pfeifer B. et al. Single-Beat Noninvasive Imaging of Cardiac Electrophysiology of Ventricular Pre-Excitation//J. Am. Coll. Cardiol., 2006; 48: 2045-2052.
This technique was applied for revealing APW-localization at manifested WPW syndrome, ectopic sources at ventricular extrasystole and tachycardia, reconstruction of the dynamics of myocardium activation at atrium flutter.
Application of the boundary element method for solving IP ECG is rather promising, in particular, in connection with that a representation of boundary conditions containing normal derivatives and a computation of normal derivatives of solution at boundary surfaces do not require numerical differentiation. However, methods above-considered have a number of disadvantages associated with specific character of the boundary element method.
The accuracy of approximating a system of boundary integral equations to a system of matrix equations directly depends on a number of elements of a boundary-element grid, as well as is rather sensitive to the quality of grid constructing. In the boundary element method a matrix of final SLAE turns out to be filled. Inversion of matrices with such structure requires considerable computational resources. In this connection, at the current level of development of mass computer techniques it is to be used matrices of a relatively not high size, i.e., to be confined by grids with a relatively small (1·103-5·103) number of elements. This circumstance puts limitations on the accuracy of solving IP ECG. A high demand of the boundary element method to the quality of grids considerably complicates a problem of automatic constructing grids on CT or MRT data.
Above-considered difficulties which already take place, when solving IP ECG for a homogeneous model of the chest, are still more growing when one makes an attempt to take into account an electrical inhomogeneity of chest tissues.
Solution of the inverse problem of electrocardiography by the boundary element method for a model of the chest with a variable coefficient of electroconductivity encounters serious mathematical difficulties.
When solving the inverse problem of electrocardiography by the boundary element method for a model of the chest with a piecewise-constant coefficient of electroconductivity, a system of 2N+1 matrix-vector equations arises, wherein N is a number of regions with different electroconductivity. Direct concatenation of a block matrix of this system in a united matrix leads to the formation of the matrix of great size with a high conditionality number. For its inversion, considerable computational resources (memory, computer fast-action) are required, and it cannot be implemented with satisfactory accuracy. Reducing a system of matrix-vector equations to a system of linear algebraic equations regarding an unknown vector of potentials by the rearrangement of a block matrix to a diagonal form requires a great number of matrix algebraic operations and, therefore, is also distinguished by low accuracy and by the necessity in considerable computational resources.
The present invention is aimed at overcoming above-mentioned disadvantages.